Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61874 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3554 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,24)(22,30)(23,28)(25,27)(26,31)(29,33)(32,34);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,24,20)(22,29,32,26)(23,28,25,27)(30,31,34,33);(19,22,24,32)(20,26,21,29)(23,31,25,33)(27,30,28,34);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,23,24,25)(20,27,21,28)(22,31,32,33)(26,30,29,34);(4,10)(8,11)(9,18)(16,17)(20,21)(26,29)(27,28)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485468);